Poisson process

economic-dictionary

The Poisson process is a time series built from experiments whose frequency can be satisfactorily approximated to a Bernoulli distribution and depends on a constant parameter called intensity.

In other words, the Poisson process is a sequence of experiments that follow a Bernoulli distribution and depends on a parameter that indicates the intensity of the process.

The time series is involved because the Poisson distribution aims to model the frequency of events during a fixed time interval.

Since the base is a Bernoulli distribution, a distinction is made between success and no success. Here it is defined success when the event that we want to control occurs and no success when it doesn't happen.

Parameter

The Greek letter “lambda” is used to identify the intensity or arrival rate of the Poisson process.

Lambda

This parameter is constant and strictly positive, that is, always greater than zero.

Formula

Given a time interval of length, t, and the rate of arrival of the events, lambda, the expected number of events during that time interval is

Expected number of events at time t

Assumptions

For the Poisson process to be feasible, the following assumptions must be met:

  1. The probability of success over a very small period of time is the lambda parameter multiplied by that period of time.
First assumption of the Poisson process
  1. The probability of more than one success event occurring in the set time interval is not significant.

In other words, the probability that more than one experiment will be successful in a fixed time interval is very small, and therefore not important or not significant.

  1. The probability of a success event occurring during a set time interval does not depend on what has happened previously.

That is, each successful experiment is independent of the previous experiment. For example, in the case of flipping a coin for 1 minute, the probability that it will come up heads does not depend on what was thrown on the previous flip.

App

The Poisson process is known in statistics as a stochastic process that aims to record very unlikely events in continuous time.

For example, in the insurance field, the Poisson process can be used to calculate the probability of ruin of an insurance company.

Poisson process example

We suppose that we want to calculate the total number of sailboats that go fishing in half an hour. We know that, on average, 4 sailboats leave every 5 minutes.

Sailboat

So, we can match the following:

Poisson process example

The expected number of sailboats that will go fishing in half an hour will be:

Poisson process example

24 sailboats will go fishing in total for half an hour taking into account that 4 sailboats are expected to go out every 5 minutes.

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